# Metrics in the sphere of a C*-module

Esteban Andruchow; Alejandro Varela

Open Mathematics (2007)

- Volume: 5, Issue: 4, page 639-653
- ISSN: 2391-5455

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topEsteban Andruchow, and Alejandro Varela. "Metrics in the sphere of a C*-module." Open Mathematics 5.4 (2007): 639-653. <http://eudml.org/doc/269531>.

@article{EstebanAndruchow2007,

abstract = {Given a unital C*-algebra \[\mathcal \{A\}\]
and a right C*-module \[\mathcal \{X\}\]
over \[\mathcal \{A\}\]
, we consider the problem of finding short smooth curves in the sphere \[\mathcal \{S\}\_\mathcal \{X\} \]
= x ∈ \[\mathcal \{X\}\]
: 〈x, x〉 = 1. Curves in \[\mathcal \{S\}\_\mathcal \{X\} \]
are measured considering the Finsler metric which consists of the norm of \[\mathcal \{X\}\]
at each tangent space of \[\mathcal \{S\}\_\mathcal \{X\} \]
. The initial value problem is solved, for the case when \[\mathcal \{A\}\]
is a von Neumann algebra and \[\mathcal \{X\}\]
is selfdual: for any element x 0 ∈ \[\mathcal \{S\}\_\mathcal \{X\} \]
and any tangent vector ν at x 0, there exists a curve γ(t) = e tZ(x 0), Z ∈ \[\mathcal \{L\}\_\mathcal \{A\} (\mathcal \{X\})\]
, Z* = −Z and ∥Z∥ ≤ π, such that γ(0) = x 0 and \[\dot\{\gamma \}\]
(0) = ν, which is minimizing along its path for t ∈ [0, 1]. The existence of such Z is linked to the extension problem of selfadjoint operators. Such minimal curves need not be unique. Also we consider the boundary value problem: given x 0, x 1 ∈ \[\mathcal \{S\}\_\mathcal \{X\} \]
, find a curve of minimal length which joins them. We give several partial answers to this question. For instance, let us denote by f 0 the selfadjoint projection I − x 0 ⊗ x 0, if the algebra f 0 \[\mathcal \{L\}\_\mathcal \{A\} (\mathcal \{X\})\]
f 0 is finite dimensional, then there exists a curve γ joining x 0 and x 1, which is minimizing along its path.},

author = {Esteban Andruchow, Alejandro Varela},

journal = {Open Mathematics},

keywords = {C*-modules; spheres; geodesics; -modules},

language = {eng},

number = {4},

pages = {639-653},

title = {Metrics in the sphere of a C*-module},

url = {http://eudml.org/doc/269531},

volume = {5},

year = {2007},

}

TY - JOUR

AU - Esteban Andruchow

AU - Alejandro Varela

TI - Metrics in the sphere of a C*-module

JO - Open Mathematics

PY - 2007

VL - 5

IS - 4

SP - 639

EP - 653

AB - Given a unital C*-algebra \[\mathcal {A}\]
and a right C*-module \[\mathcal {X}\]
over \[\mathcal {A}\]
, we consider the problem of finding short smooth curves in the sphere \[\mathcal {S}_\mathcal {X} \]
= x ∈ \[\mathcal {X}\]
: 〈x, x〉 = 1. Curves in \[\mathcal {S}_\mathcal {X} \]
are measured considering the Finsler metric which consists of the norm of \[\mathcal {X}\]
at each tangent space of \[\mathcal {S}_\mathcal {X} \]
. The initial value problem is solved, for the case when \[\mathcal {A}\]
is a von Neumann algebra and \[\mathcal {X}\]
is selfdual: for any element x 0 ∈ \[\mathcal {S}_\mathcal {X} \]
and any tangent vector ν at x 0, there exists a curve γ(t) = e tZ(x 0), Z ∈ \[\mathcal {L}_\mathcal {A} (\mathcal {X})\]
, Z* = −Z and ∥Z∥ ≤ π, such that γ(0) = x 0 and \[\dot{\gamma }\]
(0) = ν, which is minimizing along its path for t ∈ [0, 1]. The existence of such Z is linked to the extension problem of selfadjoint operators. Such minimal curves need not be unique. Also we consider the boundary value problem: given x 0, x 1 ∈ \[\mathcal {S}_\mathcal {X} \]
, find a curve of minimal length which joins them. We give several partial answers to this question. For instance, let us denote by f 0 the selfadjoint projection I − x 0 ⊗ x 0, if the algebra f 0 \[\mathcal {L}_\mathcal {A} (\mathcal {X})\]
f 0 is finite dimensional, then there exists a curve γ joining x 0 and x 1, which is minimizing along its path.

LA - eng

KW - C*-modules; spheres; geodesics; -modules

UR - http://eudml.org/doc/269531

ER -

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